This document presents an efficient convex hull algorithm for finding the convex hull of a planar set of points. The algorithm partitions the set of points into four regions based on the minimum and maximum x and y coordinates. It then finds the convex hull parts belonging to each region in parallel. Those parts are merged to derive the full convex hull. For each region, the algorithm uses a modified gradient concept to efficiently process the points and find the boundary points of the convex hull part. The algorithm achieves parallelism, data reduction, and has lower computational cost compared to traditional interior points algorithms. However, its drawbacks include difficulty extending it to higher dimensions and its static nature which requires the entire dataset from the beginning.

1. An Efficient Convex Hull Algorithm
for a Planer Set of Points
Kasun Ranga Wijeweera
(krw198708298@gmail.com)

2. This Presentation is Based on
Following Research Paper
K. R. Wijeweera, U. A. J. Pinidiyaarachchi (2013), An
Efficient Convex Hull Algorithm for a Planer Set of Points,
Ceylon Journal of Science (Physical Sciences), Volume 17, pp.
9-17.

3. Definitions
(Convex Set)
• A set S is convex if x in S and y in S implies that the segment
xy is a subset of S
• Example in 2D:

4. Definitions
(Convex Hull of a Set of Points)
• The convex hull of a set S of points is the smallest convex set
containing all the points in S
• Example in 2D:

5. A 2D Convex Hull Algorithm
(Introduction)
• This algorithm is based on gradient in coordinate geometry
• Achieves parallelism
• Achieves data reduction
• Manages coincident points
• Deals with collinear points
• Outputs hull points in boundary traversal order
• Low computational cost

6. A 2D Convex Hull Algorithm
(Inputs and Outputs)
• Input: A non-empty set of points
S = {P1, P2, . . . , Pn}
• Output: A non-empty set that contains all the points of the
convex hull of S
H = {Q1, Q2, . . . , Qm}

7. A 2D Convex Hull Algorithm
(Example Data Set)
x
y

8. A 2D Convex Hull Algorithm
(Step 1)
Calculate following values from the set S
minx // minimum value of x-coordinates
y_minx // y-coordinate corresponding to minx
maxx // maximum value of x-coordinates
y_maxx // y-coordinate corresponding to maxx
miny // minimum value of y-coordinates
x_miny // x-coordinate corresponding to miny
maxy // maximum value of y-coordinates
x_maxy // x-coordinate corresponding to maxy

9. A 2D Convex Hull Algorithm
(Basic Four Points on the Convex Hull)
(x_maxy, maxy)
(minx, y_minx)
(maxx, y_maxx)
(x_miny, miny)

10. A 2D Convex Hull Algorithm
(Step 2)
Define following subsets of S
A = { ( x, y) in S | ( x <= x_maxy ) AND ( y >= y_minx )}
B = { ( x, y) in S | ( x >= x_maxy ) AND ( y >= y_maxx )}
C = { ( x, y) in S | ( x >= x_miny ) AND ( y <= y_maxx )}
D = { ( x, y) in S | ( x <= x_miny ) AND ( y <= y_minx )}

11. A 2D Convex Hull Algorithm
(Partitioning the Data Set into Four Regions)
A B
CD

12. A 2D Convex Hull Algorithm
(Step 3)
• Find the convex hull parts belong to each of the sets A, B, C,
and D in parallel
• Merge those hull parts to derive the set H

13. A 2D Convex Hull Algorithm
(After Processing and Merging Hull Parts)
A B
CD

14. A 2D Convex Hull Algorithm
(Processing Each Set: Gradient)
x
y
M (mx, my)
N (nx, ny)
m = ( ny – my ) / ( nx – mx )

15. A 2D Convex Hull Algorithm
(Processing Each Set: Modified Gradient)
A B
CD

16. A 2D Convex Hull Algorithm
(Processing Each Set: Modified Gradient)
Region Diagram Modified Gradient Begin Point End Point
A ( ay[i] – hully ) / ( ax[i] – hullx ) x = minx
y = y_minx
x = x_maxy
y = maxy
B ( bx[i] – hullx ) / (hully - by[i] ) x = x_maxy
y = maxy
x = maxx
y = y_maxx
C ( hully - cy[i] ) / ( hullx - cx[i] ) x = maxx
y = y_maxx
x = x_miny
y = miny
D ( hullx – dx[i] ) / ( dy[i] – hully ) x = x_miny
y = miny
x = minx
y =y_minx

17. A 2D Convex Hull Algorithm
(Processing Set A: General Case)

18. A 2D Convex Hull Algorithm
(Processing Set A: General Case)

19. A 2D Convex Hull Algorithm
(Processing Set A: General Case)

20. A 2D Convex Hull Algorithm
(Processing Set A: General Case)

21. A 2D Convex Hull Algorithm
(Processing Set A: General Case)

22. A 2D Convex Hull Algorithm
(Processing Set A: Collinear Case)

23. A 2D Convex Hull Algorithm
(Processing Set A: Indeterminate Case)

24. A 2D Convex Hull Algorithm
(Processing Sets B, C, and D)
• Using similar method B, C, and D sets can also be processed
• Using the same algorithm used for A to process B, C, and D
needs additional computational cost
• Therefore four separate algorithms are used to process each of
four sets with modified gradient

25. A 2D Convex Hull Algorithm
(Interior Points Algorithm)
Based on the following Lemma
A point is non-extreme if and only if it is inside some (closed)
triangle whose vertices are points of the set and is not itself a
corner of that triangle

26. A 2D Convex Hull Algorithm
(Interior Points Algorithm)
Algorithm: INTERIOR POINTS
for each i do
for each j != i do
for each k != i != j do
for each l != k != i != j do
if p(l) in Triangle{ p(i), p(j), p(k) }
then p(l) is non-extreme

27. A 2D Convex Hull Algorithm
(Proposed Algorithm vs. Interior Points Algorithm)
• Both of the algorithms were implemented using C++
programming language
• Following hardware and software resources were used
– Computer: Intel(R) Pentium(R) Dual CPU; E2180 @ 2.00 GHz; 2.00
GHz, 0.98 GB of RAM;
– IDE: Turbo C++; Version 3.0; Copyright(c) 1990, 1992 by Borland
International, Inc;

28. A 2D Convex Hull Algorithm
(Proposed Algorithm vs. Interior Points Algorithm)
• First, n number of random points were generated in the range 0
– 399 using randomize() function and written them to a text
file
• Such text files were generated for different n
• Number of clock cycles taken to process each data file k times
was counted using clock() function

29. A 2D Convex Hull Algorithm
(Proposed Algorithm vs. Interior Points Algorithm)
Case Proposed Algorithm
(1000)
Interior Points
Algorithm (1000)
Average Ratio
(IPA : PA)
n = 10; k = 32000; 5.53 37.11 6.71
n = 20; k = 1000; 10.60 693.80 65.45
n = 30; k = 500; 15.20 3881.6 255.36
n = 40; k = 100; 17.00 12769.00 751.11
n = 50; k = 50; 24.00 32554.00 1356.41

30. A 2D Convex Hull Algorithm
(Proposed Algorithm vs. Interior Points Algorithm)
0
5000
10000
15000
20000
25000
30000
35000
Case 1 Case 2 Case 3 Case 4 Case 5
A(1000)
B(1000)

31. A 2D Convex Hull Algorithm
(Drawbacks of the Proposed Algorithm)
• It cannot be easily extended to higher dimensions
• The notion of gradient causes the problem
• Need to solve partial differential equations in order to derive
the gradient
• Because of this higher dimensional extension is not cost
effective
• The entire data set is needed from the beginning
– The algorithm is static

32. Any Questions?

33. Thank You!